On a functional equation of Luce

Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology.     

Energy of the Photosubstitutionally Reactive Excited State of Pentaammine(pyridine)ruthenium(II)

The photosensitized aquation of pentaammine(pyridine)ruthenium(II) by several dyes has been studied under conditions where only the sensitizers absorb light. The ratio of the quantum yields for ammine and pyridine substitution was the same as that for direct photoaquation. Sensitization was effective with singlet sensitizers Rhodamine-B (17 452 cm(-)(1)) and Safranine-T (17 690 cm(-)(1)), as well as the triplet sensitizer biacetyl (19 000 cm(-)(1)), but no reaction was observed with Neutral-Red (16 900 cm(-)(1)). The results indicate that the excited state precursor of the observed photosubstitution in the complex lies in the energy range between 17 000 and 17 700 cm(-)(1).     

Organopolysiloxanes as Chemically Sensitive Coatings for Optical Fibers

Various types of UV-curable organically modified siloxanes have been synthesized by the sol-gel method with the aim of fabricating chemically sensitive coatings for silica optical fibers. The refractive index of the coating material can be tailored in the range from 1.46 to 1.56 and sensitivity towards CO2 is achieved by incorporation of amino groups. The interaction of the cured layers with CO2 or with hydrocarbons has been studied in immersion experiments. Both the reaction of CO2 with incorporated amino groups and the penetration of hydrocarbons into the layer induce changes of the light absorption coefficient and the refractive index of the coating which are detected by measuring the output light intensity from the fiber.